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The route to turbulence in pipe flow is a complex, nonlinear, spatiotemporal process for which an increasingly clear understanding has emerged in recent years. This paper presents a theoretical perspective on the problem, focusing on what can be understood from relatively few physical features and models that encompass these features. The paper proceeds step-by-step with increasing detail about the transition process, first discussing the relationship to phase transitions and then exploiting an even deeper connection between pipe flow and excitable and bistable media. In the end a picture emerges for all stages of the transition process, from transient turbulence, to the onset of sustained turbulence in a percolation transition, to the modest and then rapid expansion of turbulence, ultimately leading to fully turbulent pipe flow.

We show that suitable initial disturbances to steady or long-period pulsatile flows in a straight tube with an axisymmetric 75%-occlusion stenosis can produce very large transient energy growths. The global optimal disturbances to an initially axisymmetric state found by linear analyses are three-dimensional wave packets that produce localized sinuous convective instability in extended shear layers. In pulsatile flow, initial conditions that trigger the largest disturbances are either initiated at, or advect to, the separating shear layer at the stenosis in phase with peak systolic flow. Movies are available with the online version of the paper.

Transient energy growths of two- and three-dimensional optimal linear perturbations to two-dimensional flow in a rectangular backward-facing-step geometry with expansion ratio two are presented. Reynolds numbers based on the step height and peak inflow speed are considered in the range 0–500, which is below the value for the onset of three-dimensional asymptotic instability. As is well known, the flow has a strong local convective instability, and the maximum linear transient energy growth values computed here are of order 80×103 at Re = 500. The critical Reynolds number below which there is no growth over any time interval is determined to be Re = 57.7 in the two-dimensional case. The centroidal location of the energy distribution for maximum transient growth is typically downstream of all the stagnation/reattachment points of the steady base flow. Sub-optimal transient modes are also computed and discussed. A direct study of weakly nonlinear effects demonstrates that nonlinearity is stablizing at Re = 500. The optimal three-dimensional disturbances have spanwise wavelength of order ten step heights. Though they have slightly larger growths than two-dimensional cases, they are broadly similar in character. When the inflow of the full nonlinear system is perturbed with white noise, narrowband random velocity perturbations are observed in the downstream channel at locations corresponding to maximum linear transient growth. The centre frequency of this response matches that computed from the streamwise wavelength and mean advection speed of the predicted optimal disturbance. Linkage between the response of the driven flow and the optimal disturbance is further demonstrated by a partition of response energy into velocity components.

Results are reported from a three-dimensional computational stability analysis of flow
over a backward-facing step with an expansion ratio (outlet to inlet height) of 2 at
Reynolds numbers between 450 and 1050. The analysis shows that the first absolute
linear instability of the steady two-dimensional flow is a steady three-dimensional
bifurcation at a critical Reynolds number of 748. The critical eigenmode is localized
to the primary separation bubble and has a flat roll structure with a spanwise
wavelength of 6.9 step heights. The system is further shown to be absolutely stable
to two-dimensional perturbations up to a Reynolds number of 1500. Stability spectra
and visualizations of the global modes of the system are presented for representative
Reynolds numbers.

The objective of our paper is to demonstrate the use of spectral/hp element technology in unravelling global flow instability mechanisms. Understanding these mechanisms is central to devising flow control approaches based on theoretically-founded physical principles. Global instability theory is concerned with prediction and control of linear and nonlinear disturbances developing in flows that are inhomogeneous in more than one spatial direction. As such, this theory encompasses the classic analysis of Tollmien which is valid for simple geometries of academic interest (e.g. a flat-plate) and therefore broadens the scope of this well-established but simplified methodology to include realistic problems encountered in aeronautical engineering. Compared with a direct numerical simulation approach, global instability theory can be used to explore efficiently far wider parameters ranges and deliver physical information to be used as handle for effective flow control.

The objective of our paper is to demonstrate the use of spectral/hp element technology in unravelling global flow instability mechanisms. Understanding these mechanisms is central to devising flow control approaches based on theoretically-founded physical principles. Global instability theory is concerned with prediction and control of linear and nonlinear disturbances developing in flows that are inhomogeneous in more than one spatial direction. As such, this theory encompasses the classic analysis of Tollmien which is valid for simple geometries of academic interest (e.g. a flat-plate) and therefore broadens the scope of this well-established but simplified methodology to include realistic problems encountered in aeronautical engineering. Compared with a direct numerical simulation approach, global instability theory can be used to explore efficiently far wider parameters ranges and deliver physical information to be used as handle for effective flow control.

Results are reported from a highly accurate, global numerical stability analysis of the periodic wake of a circular cylinder for Reynolds numbers between 140 and 300. The analysis shows that the two-dimensional wake becomes (absolutely) linearly unstable to three-dimensional perturbations at a critical Reynolds number of 188.5±1.0. The critical spanwise wavelength is 3.96 ± 0.02 diameters and the critical Floquet mode corresponds to a ‘Mode A’ instability. At Reynolds number 259 the two-dimensional wake becomes linearly unstable to a second branch of modes with wavelength 0.822 diameters at onset. Stability spectra and corresponding neutral stability curves are presented for Reynolds numbers up to 300.

In developed countries, most dementia appears to be due to Alzheimer's disease and vascular dementia. We report rates for incidence of subtypes of dementia based on clinical diagnosis.

Method

This study was a 2.4-year (s.d. 2.6 months) follow-up of a cohort aged 75 years and over, seen initially in a prevalence study of dementia. A screening interview in 1173 survivors was followed in a subsample of 461 respondents by a diagnostic interview 1.8 months after screening (s.d. 1.5 months). This comprised a standardised interview with respondent and informant, with venepuncture where possible. Clinical diagnoses of subtypes were made by specified criteria.

Results

The incidence of Alzheimer's disease of mild and greater severity was 2.7/1000 person-years at risk (1.6–4.4); in men 1.5 (0.8–2.7) and in women 3.3 (1.8–5.9). The incidence of vascular dementia was 1.2/100 person-years at risk (0.7–1.9); in men 1.1 (0.4–2.8) and in women 1.2 (0.7–2.0). Alzheimer's disease, but not vascular dementia, showed a marked increase with age, particularly in women. Rates for minimal dementia of different subtypes showed similar age and sex effects, but were much higher for Alzheimer's disease than vascular dementia.

Conclusions

The striking rise in incidence rates of dementia in the very old appear to be due to Alzheimer's disease, while rates for vascular dementia remain relatively constant. These trends are particularly marked for minimal dementia, but emphasise the importance of Alzheimer's disease in the community as a cause of cognitive decline of all degrees.